We address the problem of quantifying the non-markovian character of quantum time evolutions of general systems in contact with an environment. We introduce two different measures of non-markovianity that exploit the specific traits of quantum correlations and are suitable for opposite experimental contexts. When complete tomographic knowledge about the evolution is available, our measure provides a necessary and sufficient condition to quantify strictly the non-markovianity. In the opposite case, when no information whatsoever is available, we propose a sufficient condition for non-markovianity. Remarkably, no optimization procedure underlies our derivation, which greatly enhances the practical relevance of the proposed criteria.

We present a comprehensive and up-to-date review of the concept of quantum non-Markovianity, a central theme in the theory of open quantum systems. We introduce the concept of a quantum Markovian process as a generalization of the classical definition of Markovianity via the so-called divisibility property and relate this notion to the intuitive idea that links non-Markovianity with the persistence of memory effects. A detailed comparison with other definitions presented in the literature is provided. We then discuss several existing proposals to quantify the degree of non-Markovianity of quantum dynamics and to witness non-Markovian behavior, the latter providing sufficient conditions to detect deviations from strict Markovianity. Finally, we conclude by enumerating some timely open problems in the field and provide an outlook on possible research directions.

Transport phenomena are fundamental in Physics. They allow for information and energy to be exchanged between individual constituents of communication systems, networks or even biological entities. Environmental noise will generally hinder the efficiency of the transport process. However, and contrary to intuition, there are situations in classical systems where thermal fluctuations are actually instrumental in assisting transport phenomena. Here we show that, even at zero temperature, transport of excitations across dissipative quantum networks can be enhanced by local dephasing noise. We explain the underlying physical mechanisms behind this phenomenon, show that entanglement does not play a supportive role and propose possible experimental demonstrations in quantum optics. We argue that Nature may be routinely exploiting this effect and show that the transport of excitations in light harvesting molecules does benefit from such noise assisted processes. These results point towards the possibility for designing optimized structures for transport, for example in artificial nano-structures, assisted by noise.

The optimal precision of frequency measurements in the presence of decoherence is discussed. We analyze different preparations of n two-level systems as well as different measurement procedures. We show that standard Ramsey spectroscopy on uncorrelated atoms and optimal measurements on maximally entangled states provide the same resolution. The best resolution is achieved using partially entangled preparations with a high degree of symmetry. [S0031-9007(97) The rapid development of laser cooling and trapping techniques has opened up new perspectives in high precision spectroscopy. Frequency standards based on laser cooled ions are expected to achieve accuracies of the order of 1 part in 10 14 10 18 [1]. In this Letter we discuss the limits to the maximum precision achievable in the spectroscopy of n two-level atoms in the presence of decoherence. This question is particularly timely in view of current efforts to improve high precision spectroscopy by means of quantum entanglement.In the present context standard Ramsey spectroscopy refers to the situation schematically depicted in Fig. 1. An ion trap is loaded with n ions initially prepared in the same internal state j0͘. A Ramsey pulse of frequency v is applied to all ions. The pulse shape and duration are carefully chosen so that it drives the atomic transition j0͘ $ j1͘ of natural frequency v 0 and prepares an equally weighted superposition of the two internal states j0͘ and j1͘ for each ion. Next the system evolves freely for a time t followed by the second Ramsey pulse. Finally, the internal state of each particle is measured. Provided that the duration of the Ramsey pulses is much smaller than the free evolution time t, the probability that an ion is found in j1͘ is given by P ͑1 1 cos Dt͒͞2 .(1) Here D v 2 v 0 denotes the detuning between the classical driving field and the atomic transition.This basic scheme is repeated yielding a total duration T of the experiment. The aim is to estimate D as accurately as possible for a given T and a given number of ions n. The two quantities T and n are the physical resources we consider when comparing the performance of different schemes. The statistical fluctuations associated with a finite sample yield an uncertainty DP in the estimated value of P given bywhere N nT ͞t denotes the actual number of experimental data (we assume that N is large). Hence the uncertainty in the estimated value of v 0 is given byThis value is often referred to as the shot noise limit [2]. The theoretical possibility of overcoming this limit has been put forward recently [3,4]. The basic idea is to prepare the ions initially in an entangled state, which for small n seems to be practical in the near future. To see the advantage of this approach, let us consider the case of two ions prepared in the maximally entangled state [5] jC͘ ͑j00͘ 1 j11͒͘͞ p 2 .This state can be generated, for example, by the initial part of the network illustrated in Fig. 2. A Ramsey pulse on the first ion is followed by a "controlled-NOT" gate [6]. After a free evolu...